Math Problem Statement

Calculate vector sums where vectors are given in Newtons. Specific tasks include calculating F1 + F2, F3 + F4, and F5 + F6 using provided components or magnitudes and angles.

Solution

The task involves calculating vector sums where the vectors are given in newtons. Here's a breakdown of each part of the task:

a) F1 + F2:

  • Given vectors in cells A1:A4, representing x and y components of F1 and F2.
  • Follow these steps to calculate:
    1. Calculate the x-component: F1_x+F2_x=4.385+7.799=12.1849F1\_x + F2\_x = 4.385 + 7.799 = 12.1849
    2. Calculate the y-component: F1_y+F2_y=0+0=0F1\_y + F2\_y = 0 + 0 = 0
    3. Resultant vector magnitude: (12.1849)2+02=12.1849\sqrt{(12.1849)^2 + 0^2} = 12.1849
    4. Direction: Since y-component is 0, angle is 0 degrees.

b) F3 + F4:

  • Given vectors in cells A5:A8, representing x and y components of F3 and F4.
  • Follow these steps to calculate:
    1. Calculate the x-component: F3_x+F4_x=4.533+0.492=5.025F3\_x + F4\_x = -4.533 + -0.492 = -5.025
    2. Calculate the y-component: F3_y+F4_y=0+5.982=5.982F3\_y + F4\_y = 0 + 5.982 = 5.982
    3. Resultant vector magnitude: (5.025)2+(5.982)27.85\sqrt{(-5.025)^2 + (5.982)^2} \approx 7.85
    4. Direction: atan2(5.982,5.025)129.8\text{atan2}(5.982, -5.025) \approx 129.8 degrees.

c) F5 + F6:

  • Given in A9:A12 as magnitude and direction.
  • Convert to components:
    • F5: Magnitude 67.169, angle 5.67 degrees
      • x1=67.169cos(5.67)x_1 = 67.169 \cdot \cos(5.67)
      • y1=67.169sin(5.67)y_1 = 67.169 \cdot \sin(5.67)
    • F6: Magnitude 111.128, angle 111.13 degrees
      • x2=111.128cos(111.13)x_2 = 111.128 \cdot \cos(111.13)
      • y2=111.128sin(111.13)y_2 = 111.128 \cdot \sin(111.13)
  • Calculate the resultant components:
    1. x=x1+x2x = x_1 + x_2
    2. y=y1+y2y = y_1 + y_2
    3. Resultant vector magnitude: x2+y2\sqrt{x^2 + y^2}
    4. Direction: atan2(y,x)\text{atan2}(y, x) degrees.

Make sure to adjust angles to fit within the range -180 to 180 degrees when necessary.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Vector Addition
Trigonometry
Magnitude and Direction
Pythagorean Theorem
Angle Conversion

Formulas

Resultant magnitude: sqrt((x1 + x2)^2 + (y1 + y2)^2)
Resultant direction: atan2(y, x)
x-component: magnitude * cos(angle)
y-component: magnitude * sin(angle)

Theorems

Pythagorean Theorem
Trigonometric Angle Calculation

Suitable Grade Level

Undergraduate level (Physics or Mathematics)